Subgroups — #14580418

Order 2624.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₂₈	C₂	1	1	yes	maximal, characteristic
2	#13050126	C₂	1	2	yes	maximal, complemented
3	#13050084	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₄₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#107×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆₄	C₂²	1	2	yes	—
7	C₂×C₄×D₄₁	—	2	2	no	—
8	C₂×C₈×C₄₁	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
10	#10620281	—	2	2	no	—
11	C₄×Dic₄₁	C₂²	1	1	yes	characteristic
12	C₂×#66491	C₂²	1	1	yes	characteristic
13	D₃₂₈	C₄	1	4	yes	complemented
14	C₂²×D₄₁	—	2	2	no	—
15	C₂×C₄×C₄₁	—	2	2	no	—
16	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
17	C₂×Dic₄₁	D₄	1	1	yes	characteristic
18	C₂×Dic₄₁	D₄	1	1	yes	characteristic
19	D₁₆₄	—	2	2	no	—
20	D₁₆₄	C₂×C₄	1	4	yes	complemented
21	C₄×D₄₁	—	4	4	no	—
22	C₈×C₄₁	C₂×C₄	1	2	yes	complemented
23	#66491	—	2	1	no	—
24	C₂²×C₄₁	C₂×D₄	1	1	yes	characteristic
25	C₂×D₄₁	—	2	4	no	—
26	C₂×D₄₁	—	4	2	no	—
27	C₂×D₄₁	—	4	2	no	—
28	C₄×C₄₁	—	4	2	no	—
29	C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₄₁	#41	1	1	yes	characteristic
31	Dic₄₁	—	2	1	no	—
32	Dic₄₁	D₈	1	2	yes	complemented
33	C₂×C₄₁	C₄×D₄	1	1	yes	characteristic
34	C₂×C₄₁	C₂×D₈	1	1	yes	characteristic
35	C₂×C₄₁	#135	1	1	yes	characteristic
36	D₄₁	—	4	4	no	—
37	C₄×D₈	—	41	1	no	maximal, Sylow
38	C₄₁	C₄×D₈	1	1	yes	characteristic, complemented, Sylow
39	C₂×D₈	—	41	1	no	—
40	C₄×D₄	—	41	2	no	—
41	#107	D₄₁	1	1	yes	characteristic, complemented
42	#102	—	41	2	no	—
43	C₄×C₈	—	41	1	no	—
44	C₂×D₄	—	41	2	no	—
45	C₂²×C₄	—	82	2	no	—
46	D₈	—	41	4	no	—
47	C₂×C₈	—	41	1	no	—
48	C₂×C₈	C₂×D₄₁	1	1	yes	characteristic
49	C₄⋊C₄	C₂×D₄₁	1	2	yes	—
50	C₂²⋊C₄	—	82	2	no	—
51	C₄²	—	41	1	no	—
52	C₂³	—	82	2	no	—
53	D₄	—	41	4	no	—
54	D₄	—	82	2	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	41	1	no	—
57	C₂×C₄	—	41	1	no	—
58	C₂×C₄	—	164	4	no	—
59	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
60	C₈	—	82	1	no	—
61	C₈	C₄×D₄₁	1	2	yes	complemented
62	C₂²	—	82	4	no	—
63	C₂²	—	164	2	no	—
64	C₂²	—	164	2	no	—
65	C₂²	D₄×D₄₁	1	1	yes	characteristic
66	C₄	—	4	2	no	—
67	C₄	—	41	2	no	—
68	C₄	—	82	1	no	—
69	C₄	C₂×C₄×D₄₁	1	1	yes	characteristic
70	C₄	#10620310	1	1	yes	characteristic
71	C₂	—	164	4	no	—
72	C₂	#13050126	1	1	yes	characteristic
73	C₂	D₈×D₄₁	1	1	yes	characteristic
74	C₂	#13050187	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₄₁
commutator	C₄×C₄₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₁
Fitting	#107×C₄₁	C₂
socle	C₂²×C₄₁	C₂×D₄
prime radical	C₂×D₃₂₈	C₂
cosocle	C₂×C₄×C₄₁	C₂³
hypercenter	#107	D₄₁
hypocenter	C₄₁	C₄×D₈
LCS	G ▷ C₄×C₄₁ ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #13050126	C₂	semidirect
3. #13050084	C₂	semidirect
4. C₈×Dic₄₁	C₂	semidirect
5. #107×C₄₁	C₂	semidirect
13. D₃₂₈	C₄	semidirect
20. D₁₆₄	C₂×C₄	semidirect
22. C₈×C₄₁	C₂×C₄	semidirect
32. Dic₄₁	D₈	semidirect
38. C₄₁	C₄×D₈	semidirect
41. #107	D₄₁	semidirect
61. C₈	C₄×D₄₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×D₃₂₈	2
#13050126	2
#13050084	2
C₈×Dic₄₁	2
#107×C₄₁	2
C₄×D₈	41